Nonlinear Maps Satisfying Derivability on the Parabolic Subalgebras of the Full Matrix Algebras

DOI：10.3770/j.issn:1000-341X.2011.05.004

 作者 单位 陈正新 福建师范大学数学与计算机科学学院, 福建 福州 350007 赵玉娥 青岛大学数学科学学院, 山东 青岛 266071

设$F$为特征为零的域, $M_n(\mathbb{F})$ 为$\mathbb{F}$上$n$ 阶全矩阵代数, $n\geq 2$, $\bf{t}$ 为 $M_n(\mathbb{F})$中所有上三角矩阵代数组成的子代数.称$M_n(\mathbb{F})$中包含$\bf{t}$ 的子代数$\bf{P}$ 为$M_n(\mathbb{F})$ 的抛物子代数.$M_n(\mathbb{F})$的抛物子代数$\bf{P}$ 上一个映射$\varphi$ 称为具有可导性,如果对任意的$A,B\in \bf{P}$, $\varphi(AB)=\varphi (A)B A\varphi(B)$.这里的$\varphi$ 不一定为线性映射. 本文证明$\bf{P}$上一个映射$\varphi$具有可导性当且仅当$\varphi$ 是$\bf{P}$ 上内导子和加法拟导子的和.

Let ${\mathbb{F}}$ be a field of characteristic $0$, $M_n({\mathbb{F}})$ the full matrix algebra over ${\mathbb{F}}$, ${\bf t}$ the subalgebra of $M_n({\mathbb{F}})$ consisting of all upper triangular matrices. Any subalgebra of $M_n({\mathbb{F}})$ containing ${\bf t}$ is called a parabolic subalgebra of $M_n({\mathbb{F}})$. Let ${\bf P}$ be a parabolic subalgebra of $M_n({\mathbb{F}})$. A map $\varphi$ on ${\bf P}$ is said to satisfy derivability if $\varphi (x\cdot y)=\varphi (x)\cdot y x\cdot \varphi(y)$ for all $x,y\in {\bf P}$, where $\varphi$ is not necessarily linear. Note that a map satisfying derivability on ${\bf P}$ is not necessarily a derivation on ${\bf P}$. In this paper, we prove that a map $\varphi$ on ${\bf P}$ satisfies derivability if and only if $\varphi$ is a sum of an inner derivation and an additive quasi-derivation on ${\bf P}$. In particular, any derivation of parabolic subalgebras of $M_n({\mathbb{F}})$ is an inner derivation.